Computes probability density (mass) functions
Category: Probability
Alias: PMF
PDF ( 'dist',quantile,parm-1, ,parm-k )
'dist'
is a character string that identifies the distribution. Valid distributions are as follows :
Distribution | Argument |
---|---|
Bernoulli | 'BERNOULLI' |
Beta | 'BETA' |
Binomial | 'BINOMIAL' |
Cauchy | 'CAUCHY' |
Chi-Square | 'CHISQUARE' |
Exponential | 'EXPONENTIAL' |
F | 'F' |
Gamma | 'GAMMA' |
Geometric | 'GEOMETRIC' |
Hypergeometric | 'HYPERGEOMETRIC' |
Laplace | 'LAPLACE' |
Logistic | 'LOGISTIC' |
Lognormal | 'LOGNORMAL' |
Negative binomial | 'NEGBINOMIAL' |
Normal | 'NORMAL''GAUSS' |
Normal mixture | 'NORMALMIX' |
Pareto | 'PARETO' |
Poisson | 'POISSON' |
T | 'T' |
Uniform | 'UNIFORM' |
Wald (inverse Gaussian) | 'WALD''IGAUSS' |
Weibull | 'WEIBULL' |
Note: Except for T, F, and NORMALMIX, you can minimally identify any distribution by its first four characters .
quantile
is a numeric random variable.
parm-1, ,parm-k
are optional shape , location , or scale parameters that are appropriate for the specific distribution.
See: 'Details' on page 696 for complete information about these parameters
Bernoulli Distribution
PDF ('BERNOULLI', x , p )
where
x
is a numeric random variable.
p
is a numeric probability of success.
Range: ‰ p ‰ 1
The PDF function for the Bernoulli distribution returns the probability density function of a Bernoulli distribution, with probability of success equal to p . The PDF function is evaluated at the value x . The equation follows:
Note: There are no location or scale parameters for this distribution.
Beta Distribution
PDF ('BETA', x , a , b <, l , r >)
where
x
is a numeric random variable.
a
is a numeric shape parameter.
Range: a > 0
b
is a numeric shape parameter.
Range: b > 0
l
is the numeric left location parameter.
Default:
r
is the right location parameter.
Default:
Range: r > l
The PDF function for the beta distribution returns the probability density function of a beta distribution, with shape parameters a and b . The PDF function is evaluated at the value x . The equation follows:
Note: The quantity is forced to be
Binomial Distribution
PDF ('BINOMIAL', m , p , n )
where
m
is an integer random variable that counts the number of successes.
Range: m = 0, 1,
p
is a numeric probability of success.
Range: ‰ p ‰ 1
n
is an integer parameter that counts the number of independent Bernoulli trials.
Range: n = 0, 1,
The PDF function for the binomial distribution returns the probability density function of a binomial distribution, with parameters p and n , which is evaluated at the value m . The equation follows:
Note: There are no location or scale parameters for the binomial distribution.
Cauchy Distribution
PDF ('CAUCHY', x <, , » >)
where
x
is a numeric random variable.
is a numeric location parameter.
Default:
»
is a numeric scale parameter.
Default: 1
Range: » > 0
The PDF function for the Cauchy distribution returns the probability density function of a Cauchy distribution, with the location parameter and the scale parameter » . The PDF function is evaluated at the value x . The equation follows:
Chi-Square Distribution
PDF ('CHISQUARE', x , df <, nc >)
where
x
is a numeric random variable.
df
is a numeric degrees of freedom parameter.
Range: df > 0
nc
is an optional numeric non- centrality parameter.
Range: nc ‰
The PDF function for the chi-square distribution returns the probability density function of a chi-square distribution, with df degrees of freedom and non-centrality parameter nc . The PDF function is evaluated at the value x . This function accepts non-integer degrees of freedom. If nc is omitted or equal to zero, the value returned is from the central chi-square distribution. The following equation describes the PDF function of the chi-square distribution,
where p c (.,.) denotes the density from the central chi-square distribution:
and where p g ( y , b ) is the density from the gamma distribution, which is given by
Exponential Distribution
PDF ('EXPONENTIAL', x <, » >)
where
x
is a numeric random variable.
»
is a scale parameter.
Default: 1
Range: » > 0
The PDF function for the exponential distribution returns the probability density function of an exponential distribution, with the scale parameter » . The PDF function is evaluated at the value x . The equation follows:
F Distribution
PDF ('F', x , ndf , ddf <, nc >)
where
x
is a numeric random variable.
ndf
is a numeric numerator degrees of freedom parameter.
Range: ndf > 0
ddf
is a numeric denominator degrees of freedom parameter.
Range: ddf > 0
nc
is a numeric non-centrality parameter.
Range: nc ‰
The PDF function for the F distribution returns the probability density function of an F distribution, with ndf numerator degrees of freedom, ddf denominator degrees of freedom, and non-centrality parameter nc . The PDF function is evaluated at the value x . This PDF function accepts non-integer degrees of freedom for ndf and ddf . If nc is omitted or equal to zero, the value returned is from a central F distribution. The following equation describes the PDF function of the F distribution,
where p f ( f , u 1 , u 2 ) is the density from the central F distribution with
and where p B ( x , a , b ) is the density from the standard beta distribution.
Note: There are no location or scale parameters for the F distribution.
Gamma Distribution
PDF ('GAMMA', x , a <, » >)
where
x
is a numeric random variable.
a
is a numeric shape parameter.
Range: a > 0
»
is a numeric scale parameter.
Default: 1
Range: » > 0
The PDF function for the gamma distribution returns the probability density function of a gamma distribution, with the shape parameter a and the scale parameter » . The PDF function is evaluated at the value x . The equation follows:
Geometric Distribution
PDF ('GEOMETRIC', m,p )
where
m
is a numeric random variable that denotes the number of failures.
Range: m ‰
p
is a numeric probability of success.
Range: ‰ p ‰ 1
The PDF function for the geometric distribution returns the probability density function of a geometric distribution, with parameter p . The PDF function is evaluated at the value m . The equation follows:
Note: There are no location or scale parameters for this distribution.
Hypergeometric Distribution
CDF ('HYPER', x,N,R,n <, o >)
where
x
is an integer random variable.
N
is an integer population size parameter.
Range: N = 1, 2, ...
R
is an integer number of items in the category of interest.
Range: R = 0, 1, ..., N
n
is an integer sample size parameter.
Range: n = 1, 2, ..., N
o
is an optional numeric odds ratio parameter.
Range: o > 0
The PDF function for the hypergeometric distribution returns the probability density function of an extended hypergeometric distribution, with population size N , number of items R , sample size n , and odds ratio o . The PDF function is evaluated at the value x . If o is omitted or equal to 1, the value returned is from the usual hypergeometric distribution. The equation follows:
Laplace Distribution
PDF ('LAPLACE', x <, , » >)
where
x
is a numeric random variable.
is a numeric location parameter.
Default:
»
is a numeric scale parameter.
Default: 1
Range: » > 0
The PDF function for the Laplace distribution returns the probability density function of the Laplace distribution, with the location parameter and the scale parameter » . The PDF function is evaluated at the value x . The equation follows:
Logistic Distribution
PDF ('LOGISTIC', x <, , » >)
where
x
is a numeric random variable.
is a numeric location parameter.
Default:
»
is a numeric scale parameter.
Default: 1
Range: » > 0
The PDF function for the logistic distribution returns the probability density function of a logistic distribution, with the location parameter and the scale parameter » . The PDF function is evaluated at the value x . The equation follows:
Lognormal Distribution
PDF ('LOGNORMAL', x <, , » >)
where
x
is a numeric random variable.
is a numeric location parameter.
Default:
»
is a numeric scale parameter.
Default: 1
Range: » > 0
The PDF function for the lognormal distribution returns the probability density function of a lognormal distribution, with the location parameter and the scale parameter » . The PDF function is evaluated at the value x . The equation follows:
Negative Binomial Distribution
PDF ('NEGBINOMIAL', m,p,n )
where
m
is a positive integer random variable that counts the number of failures.
Range: m = 0, 1, ...
p
is a numeric probability of success.
Range: ‰ p ‰ 1
n
is an integer parameter that counts the number of successes.
Range: n = 1, 2, ...
The PDF function for the negative binomial distribution returns the probability density function of a negative binomial distribution, with probability of success p and number of successes n . The PDF function is evaluated at the value m . The equation follows:
Note: There are no location or scale parameters for the negative binomial distribution.
Normal Distribution
PDF ('NORMAL', x <, , » >)
where
x
is a numeric random variable.
is a numeric location parameter.
Default:
»
is a numeric scale parameter.
Default: 1
Range: » > 0
The PDF function for the normal distribution returns the probability density function of a normal distribution, with the location parameter and the scale parameter » . The PDF function is evaluated at the value x . The equation follows:
Normal Mixture Distribution
CDF ('NORMALMIX', x,n,p,m,s )
where
x
is a numeric random variable.
n
is the integer number of mixtures.
Range: n = 1, 2, ...
p
is the n proportions , p 1 , p 2 ,..., p n , where .
Range: p = 0, 1, ...
m
is the n means m 1 , m 2 ,..., m n .
s
is the n standard deviations s 1 , s 2 ,..., s n .
Range: s > 0
The PDF function for the normal mixture distribution returns the probability that an observation from a mixture of normal distribution is less than or equal to x . The equation follows:
Note: There are no location or scale parameters for the normal mixture distribution.
Pareto Distribution
PDF ('PARETO', x,a <, k >)
where
x
is a numeric random variable.
a
is a numeric shape parameter.
Range: a > 0
k
is a numeric scale parameter.
Default: 1
Range: k > 0
The PDF function for the Pareto distribution returns the probability density function of a Pareto distribution, with the shape parameter a and the scale parameter k . The PDF function is evaluated at the value x . The equation follows:
Poisson Distribution
PDF ('POISSON', n,m )
where
n
is an integer random variable.
Range: n =0,1,...
m
is a numeric mean parameter.
Range: m > 0
The PDF function for the Poisson distribution returns the probability density function of a Poisson distribution, with mean m . The PDF function is evaluated at the value n . The equation follows:
Note: There are no location or scale parameters for the Poisson distribution.
T Distribution
PDF ('T', t,df <, nc >)
where
t
is a numeric random variable.
df
is a numeric degrees of freedom parameter.
Range: df > 0
nc
is an optional numeric non-centrality parameter.
The PDF function for the T distribution returns the probability density function of a T distribution, with degrees of freedom df and non-centrality parameter nc . The PDF function is evaluated at the value x . This PDF function accepts non-integer degrees of freedom. If nc is omitted or equal to zero, the value returned is from the central T distribution. The equation follows:
Note: There are no location or scale parameters for the T distribution.
Uniform Distribution
PDF ('UNIFORM', x <, l,r >)
where
x
is a numeric random variable.
l
is the numeric left location parameter.
Default:
r
is the numeric right location parameter.
Default: 1
Range: r > l
The PDF function for the uniform distribution returns the probability density function of a uniform distribution, with the left location parameter l and the right location parameter r . The PDF function is evaluated at the value x . The equation follows:
Wald (Inverse Gaussian) Distribution
PDF ('WALD', x,d )
PDF ('IGAUSS', x,d )
where
x
is a numeric random variable.
d
is a numeric shape parameter.
Range: d > 0
The PDF function for the Wald distribution returns the probability density function of a Wald distribution, with shape parameter d , which is evaluated at the value x . The equation follows:
Note: There are no location or scale parameters for the Wald distribution.
Weibull Distribution
PDF ('WEIBULL', x,a <, » >)
where
x
is a numeric random variable.
a
is a numeric shape parameter.
Range: a > 0
»
is a numeric scale parameter.
Default: 1
Range: » > 0
The PDF function for the Weibull distribution returns the probability density function of a Weibull distribution, with the shape parameter a and the scale parameter » . The PDF function is evaluated at the value x . The equation follows:
SAS Statements | Results |
---|---|
y=pdf( ' BERN ' ,0,.25); | 0.75 |
y=pdf( ' BERN ' ,1,.25); | 0.25 |
y=pdf( ' BETA ' ,0.2,3,4); | 1.2288 |
y=pdf( ' BINOM ' ,4,.5,10); | 0.20508 |
y=pdf( ' CAUCHY ' ,2); | 0.063662 |
y=pdf( ' CHISQ ' ,11.264,11); | 0.081686 |
y=pdf( ' EXPO ' ,1); | 0.36788 |
y=pdf( ' F ' ,3.32,2,3); | 0.054027 |
y=pdf( ' GAMMA ' ,1,3); | 0.18394 |
y=pdf( ' HYPER ' ,2,200,50,10); | 0.28685 |
y=pdf( ' LAPLACE ' ,1); | 0.18394 |
y=pdf( ' LOGISTIC ' ,1); | 0.19661 |
y=pdf( ' LOGNORMAL ' ,1); | 0.39894 |
y=pdf( ' NEGB ' ,1,.5,2); | 0.25 |
y=pdf( ' NORMAL ' ,1.96); | 0.058441 |
y=pdf('NORMALMIX',2.3,3,.33,.33,.34, .5,1.5,2.5,.79,1.6,4.3); | 0.1166 |
y=pdf( ' PARETO ' ,1,1); | 1 |
y=pdf( ' POISSON ' ,2,1); | 0.18394 |
y=pdf( ' T ' ,.9,5); | 0.24194 |
y=pdf( ' UNIFORM ' ,0.25); | 1 |
y=pdf( ' WALD ' ,1,2); | 0.56419 |
y=pdf( ' WEIBULL ' ,1,2); | 0.73576 |